# Effects of Convective Heating on Entropy Generation Rate in a Channel with Permeable Walls

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

_{f}which provides a heat transfer coefficient ${\gamma}_{1}$ while the upper wall losses heat to the ambient with heat transfer coefficient ${\gamma}_{2}$.

_{∞}is the ambient temperature. We introduce the following non-dimensional quantities:

## 3. Entropy Analysis

## 4. Results and Discussion

**Table 1.**Computation showing comparison between the exact and numerical solution of velocity profile for G = 1, Re = 1.

η | Exact Solutionw(η) | Numerical Solution w(η) |
---|---|---|

0 | 0 | 0 |

0.1 | 0.03879297 | 0.03879297 |

0.2 | 0.07114875 | 0.07114875 |

0.3 | 0.09639032 | 0.09639032 |

0.4 | 0.11376948 | 0.09639032 |

0.5 | 0.12245933 | 0.11376948 |

0.6 | 0.12154600 | 0.12154600 |

0.7 | 0.11001953 | 0.11001953 |

0.8 | 0.08676372 | 0.08676372 |

0.9 | 0.05054498 | 0.05054498 |

1.0 | 0 | 0 |

#### 4.1. Effects of Parameter Variations on Velocity and Temperature Profiles

#### 4.2. Effects of Parameter Variations on Entropy Generation Rate

#### 4.3. Effects of Parameter Variations on Bejan Number

## 5. Conclusions

- (1)
- The fluid temperature increases with increasing Re, Ec, Bi
_{1}, Pr and decreases with increasing values of Bi_{2}. - (2)
- Entropy generation rate increases with increasing values of Bi
_{1}, Bi_{2}, $Br{\mathrm{\Omega}}^{-1}$. As Re increases, entropy production decreases at lower wall and increases at the upper wall. - (3)
- Increase in Bi
_{1}, Bi_{2}enhance dominant effects of heat transfer irreversibility while increase in $Br{\mathrm{\Omega}}^{-1}$ enhance dominant effects of fluid friction irreversibility. - (4)
- Increase in Re decrease Bejan number at the lower wall region and increase Bejan number at the upper wall region.
- (5)
- Heat transfer irreversibility dominates the centerline region of the channel.

## Nomenclature

${C}_{P}$ | Specific heat at a constant pressure |

${N}_{2}$ | Entropy generation due to viscous dissipation |

$u$ | Fluid velocity |

$k$ | Thermal conductivity |

$V$ | Uniform suction/injection velocity |

$P$ | Fluid pressure |

${N}_{1}$ | Entropy generation due to heat transfer |

$T$ | Temperature |

${E}_{G}$ | Local volumetric rate of entropy generation |

$Be$ | Bejan number |

${T}_{f}$ | Hot fluid temperature |

${T}_{\infty}$ | Ambient temperature |

${T}_{h}$ | Temperature at suction wall |

$T(0)$ | Temperature at injection wall |

$h$ | Channel width. |

$G$ | Pressure gradient |

$\mathrm{Re}$ | Reynolds number |

$\mathrm{Pr}$ | Prandtl number |

$Br$ | Brinkman number |

$Ec$ | Eckert number |

$B{i}_{1}$ | Lower wall Biot number |

$B{i}_{2}$ | Upper wall Biot number |

$x,y$ | Cartesian coordinates |

$w$ | Dimensionless velocity |

$X$ | Dimensionless axial coordinate |

## Greek symbols

$\alpha $ | Thermal diffusivity |

$\mu $ | Fluid viscosity |

$\theta $ | Dimensionless temperature |

$\Phi $ | Irreversibility ratio |

${\gamma}_{1}$ | Lower heat transfer coefficient |

${\gamma}_{2}$ | Upper heat transfer coefficient |

$\Omega $ | Temperature difference |

$\rho $ | Fluid density |

$\eta $ | Dimensionless transverse coordinate |

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**MDPI and ACS Style**

Makinde, O.D.; Eegunjobi, A.S.
Effects of Convective Heating on Entropy Generation Rate in a Channel with Permeable Walls. *Entropy* **2013**, *15*, 220-233.
https://doi.org/10.3390/e15010220

**AMA Style**

Makinde OD, Eegunjobi AS.
Effects of Convective Heating on Entropy Generation Rate in a Channel with Permeable Walls. *Entropy*. 2013; 15(1):220-233.
https://doi.org/10.3390/e15010220

**Chicago/Turabian Style**

Makinde, Oluwole Daniel, and Adetayo Samuel Eegunjobi.
2013. "Effects of Convective Heating on Entropy Generation Rate in a Channel with Permeable Walls" *Entropy* 15, no. 1: 220-233.
https://doi.org/10.3390/e15010220